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  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "* 我们学习了深度网络是如何实现和优化的。在这一过程中只是用张量和自动微分，不需要定义层或复杂的优化器。\n",
    "* 本节只触及了表面只是。在下面的部分中，我们将基于刚才姐扫的概念描述其他类型，并学习如何更简洁地实现其他模型"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 49,
   "metadata": {},
   "outputs": [],
   "source": [
    "import os\n",
    "os.environ[\"KMP_DUPLICATE_LIB_OK\"] = \"TRUE\""
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 50,
   "metadata": {},
   "outputs": [],
   "source": [
    "%matplotlib inline\n",
    "import random\n",
    "import torch\n",
    "from d2l import torch as d2l"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# 3.2.1 生成数据集"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "生成一个包含1000个样本的数据集，每个样本包含从标准正态分布中抽样的两个特征。我们合成数据集是一个矩阵$X \\in \\mathbb{R}^{1000 \\times 2}$。"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "我们使用线性模型参数$w = [2,-3.4] ^ T$ 和$b = 4.2$和噪声项$\\epsilon$生成数据集及其标签："
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$$y = Xw + b + \\epsilon$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$\\epsilon$可以视为模型预测和标签的潜在观测误差。在这里我们认为标准假设成立，即$\\epsilon$服从均值为0的正态分布。"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 51,
   "metadata": {},
   "outputs": [],
   "source": [
    "#@save\n",
    "def synthetic_data(w, b, num_examples):\n",
    "    \"\"\"生成y=Xw+b+噪声\"\"\"\n",
    "    X = torch.normal(0, 1, (num_examples, len(w)))\n",
    "    y = torch.matmul(X, w) + b\n",
    "    y += torch.normal(0, 0.01, y.shape)\n",
    "    return X, y.reshape((-1, 1))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 52,
   "metadata": {},
   "outputs": [],
   "source": [
    "true_w = torch.tensor([2, -3.4])\n",
    "true_b = 4.2\n",
    "features, labels = synthetic_data(true_w, true_b, 1000)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 53,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "features: tensor([-1.0833,  0.2739]) \n",
      "labels: tensor([1.1141])\n"
     ]
    }
   ],
   "source": [
    "print('features:',features[0],'\\nlabels:',labels[0])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 54,
   "metadata": {},
   "outputs": [
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   ],
   "source": [
    "d2l.set_figsize()\n",
    "d2l.plt.scatter(features[:,(1)].detach().numpy(), labels.detach().numpy(),1)\n",
    "d2l.plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# 3.2.2 读取数据集"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "def data_iter(batch_size, features, labels):\n",
    "    num_examples = len(features)\n",
    "    indices = list(range(num_examples))\n",
    "    # 这些样本是随机读取的，没有特定的顺序\n",
    "    random.shuffle(indices)\n",
    "    for i in range(0, num_examples, batch_size):\n",
    "        batch_indices = torch.tensor(\n",
    "            indices[i: min(i + batch_size, num_examples)])\n",
    "        yield features[batch_indices], labels[batch_indices]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 56,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "tensor([[ 0.2858, -0.4004],\n",
      "        [ 0.1898,  1.4706],\n",
      "        [-0.5907, -1.6506],\n",
      "        [-0.9729,  0.1563],\n",
      "        [ 0.7741, -0.2239],\n",
      "        [-1.2543,  0.2823],\n",
      "        [-0.5527, -0.2908],\n",
      "        [-0.4338, -1.5134],\n",
      "        [ 1.6515, -0.2922],\n",
      "        [-1.7382, -0.2711]]) \n",
      " tensor([[ 6.1212],\n",
      "        [-0.4122],\n",
      "        [ 8.6231],\n",
      "        [ 1.7419],\n",
      "        [ 6.5051],\n",
      "        [ 0.7284],\n",
      "        [ 4.0906],\n",
      "        [ 8.4805],\n",
      "        [ 8.4839],\n",
      "        [ 1.6532]])\n"
     ]
    }
   ],
   "source": [
    "batch_size = 10\n",
    "\n",
    "for X,y in data_iter(batch_size, features, labels):\n",
    "    print(X,'\\n',y)\n",
    "    break"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# 3.2.3 初始化模型参数"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "我们通过从均值为0，标准差为0.01的正态分布中抽样随机数来初始化权重，并将偏置初始化为0。"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 57,
   "metadata": {},
   "outputs": [],
   "source": [
    "w = torch.normal(0,0.01,size = (2,1),requires_grad=True)\n",
    "b = torch.zeros(1,requires_grad=True)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# 3.2.4 定义模型"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "只需计算输入特征$X$和模型权重$w$的矩阵-向量乘法后，再加上偏差$b$并应用激活函数。"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 58,
   "metadata": {},
   "outputs": [],
   "source": [
    "#@save\n",
    "def linreg(X,w,b):\n",
    "    \"\"\"线性回归模型\"\"\"\n",
    "    return torch.matmul(X,w) + b"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# 3.2.5 定义损失函数"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "在实现中，我们需要将真实值$y$的形状转换为和预测值$\\hat y$的形状相同。"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 59,
   "metadata": {},
   "outputs": [],
   "source": [
    "#@save\n",
    "def squared_loss(y_hat,y):\n",
    "    return (y_hat-y.reshape(y_hat.shape))**2/2"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# 3.2.6 定义优化算法"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "每一步更新的大小由学习率$lr$决定。"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 61,
   "metadata": {},
   "outputs": [],
   "source": [
    "#@save\n",
    "def sgd(params, lr, batch_size):\n",
    "    \"\"\"\"小批量随机梯度下降\"\"\"\n",
    "    with torch.no_grad():\n",
    "        for param in params:\n",
    "            param -= lr * param.grad / batch_size\n",
    "            param.grad.zero_()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# 3.2.7 训练"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "在每次迭代中，我们读取小批量训练样本，并通过我们的模型来获得一组预测。计算完损失后，我们开始反向传播，存储每个参数的梯度。最后，我们调用优化算法sgd来更新模型参数。"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "1. 初始化参数\n",
    "2. 重复以下训练，直到完成：\n",
    "   * 计算梯度 $g \\leftarrow \\partial_{(w,b)} \\frac{1}{|B|} \\sum_{i \\in B} l(x_{(i)},y_{(i)}, w, b)$\n",
    "   * 更新参数 $(w,b) \\leftarrow (w,b) \\leftarrow \\eta g$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 62,
   "metadata": {},
   "outputs": [],
   "source": [
    "lr = 0.03\n",
    "num_epochs = 3\n",
    "net = linreg\n",
    "loss = squared_loss"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 63,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "epoch 1, loss 0.034311\n",
      "epoch 2, loss 0.000120\n",
      "epoch 3, loss 0.000048\n"
     ]
    }
   ],
   "source": [
    "for epoch  in range(num_epochs):\n",
    "    for X,y in data_iter(batch_size, features, labels):\n",
    "        l = loss(net(X,w,b), y)# X 和 y 的小批量损失\n",
    "        # 因为l形状是(batch_size,1)，而不是一个标量。\n",
    "        # l中的所有元素被加到一起，并以此计算关于[w,b]的梯度\n",
    "        l.sum().backward()\n",
    "        sgd([w,b], lr, batch_size) # 使用小批量随机梯度下降迭代模型参数\n",
    "    with torch.no_grad():\n",
    "        train_l = loss(net(features,w,b), labels)\n",
    "        print(f'epoch {epoch+1}, loss {float(train_l.mean()):f}')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 64,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "w 的估计误差：tensor([ 0.0001, -0.0003], grad_fn=<SubBackward0>)\n",
      "b 的估计误差：tensor([0.0009], grad_fn=<RsubBackward1>)\n"
     ]
    }
   ],
   "source": [
    "print(f'w 的估计误差：{true_w - w.reshape(true_w.shape)}')\n",
    "print(f'b 的估计误差：{true_b - b}')"
   ]
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